We investigate the electronic structure of the flat bands induced by moiré superlattices and electric fields in nearly aligned ABC trilayer graphene (TLG) boron-nitride (BN) interfaces where Coulomb effects can lead to correlated gapped phases. Our calculations indicate that valley-spin resolved isolated superlattice flat bands that carry a finite Chern number C=3 proportional to the layer number can appear near charge neutrality for appropriate perpendicular electric fields and twist angles. When the degeneracy of the bands is lifted by Coulomb interactions, these topological bands can lead to anomalous quantum Hall phases that embody orbital and spin magnetism. Narrow bandwidths of ∼10 meV achievable for a continuous range of twist angles θ≲0.6° with moderate interlayer potential differences of ∼50 meV make the TLG-BN systems a promising platform for the study of electric-field tunable Coulomb-interaction-driven spontaneous Hall phases.